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Classification of Extended Abelian Chern-Simons Theories

Daniel Galviz

Abstract

We classify extended Abelian Chern-Simons theories with gauge group $U(1)^n$ as extended $(2+1)$-dimensional topological quantum field theories. For an even integral nondegenerate lattice $(Λ,K)$, let $(G_K,q_K)$ denote its discriminant quadratic module. We prove that the associated theory is determined, up to symmetric monoidal natural isomorphism, by this finite quadratic module, and that every finite quadratic module is realized as the discriminant quadratic module of an even integral nondegenerate lattice. It follows that finite quadratic modules classify extended Abelian Chern-Simons theories, pointed Abelian Reshetikhin-Turaev TQFTs, and pointed modular tensor categories.

Classification of Extended Abelian Chern-Simons Theories

Abstract

We classify extended Abelian Chern-Simons theories with gauge group as extended -dimensional topological quantum field theories. For an even integral nondegenerate lattice , let denote its discriminant quadratic module. We prove that the associated theory is determined, up to symmetric monoidal natural isomorphism, by this finite quadratic module, and that every finite quadratic module is realized as the discriminant quadratic module of an even integral nondegenerate lattice. It follows that finite quadratic modules classify extended Abelian Chern-Simons theories, pointed Abelian Reshetikhin-Turaev TQFTs, and pointed modular tensor categories.

Paper Structure

This paper contains 3 sections, 7 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Extended Theories and Equivalence Theorems
  3. Classification by Finite Quadratic Modules

Key Result

Theorem 2.1

Galviz2 Let $\mathbb T=\mathfrak t/\Lambda\cong U(1)^n$ and let $K:\Lambda\times\Lambda\to\mathbb Z$ be even, integral, and nondegenerate. With the $K$-twisted extended bordism convention, the assignments together with the weighted extended assignment define a unitary extended $(2+1)$-dimensional topological quantum field theory. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (15)

  • Theorem 2.1
  • Remark
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Remark
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 5 more